Subgroups — #12067

Order 256.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2177	C₂	1	1	yes	maximal, characteristic
2	#1580	C₂	1	2	yes	maximal, complemented
3	#1368	C₂	1	1	yes	maximal, characteristic, complemented
4	#1333	C₂	1	2	yes	maximal, complemented
5	#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆	C₄	1	1	yes	characteristic
7	C₄×D₈	C₂²	1	2	yes	—
8	C₈×D₄	—	2	2	no	—
9	#367	C₄	1	1	yes	characteristic
10	#357	C₄	1	2	yes	—
11	#346	C₂²	1	1	yes	characteristic
12	#346	C₂²	1	1	yes	characteristic
13	#334	C₂²	1	2	yes	—
14	D₄⋊C₈	—	2	2	no	—
15	#322	C₂²	1	1	yes	characteristic
16	C₂×D₈	C₂×C₄	1	2	yes	—
17	C₂²×C₈	—	4	2	no	—
18	C₄×D₄	—	2	2	no	—
19	D₁₆	C₈	1	4	yes	complemented
20	C₂×C₁₆	—	2	1	no	—
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	#107	C₂×C₄	1	2	yes	—
24	C₄⋊C₈	—	2	2	no	—
25	#102	—	2	2	no	—
26	C₂²⋊C₈	—	4	2	no	—
27	C₄×C₈	D₄	1	2	yes	—
28	C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂×D₄	—	2	2	no	—
30	C₂²×C₄	—	4	2	no	—
31	D₈	—	2	2	no	—
32	D₈	C₂×C₈	1	4	yes	complemented
33	C₂×C₈	—	2	2	no	—
34	C₂×C₈	—	4	2	no	—
35	C₂×C₈	—	8	4	no	—
36	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₈	#41	1	1	yes	characteristic
38	C₄⋊C₄	—	2	2	no	—
39	C₂²⋊C₄	—	4	2	no	—
40	C₄²	C₂×D₄	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	—	4	1	no	—
43	C₁₆	C₂×C₈	1	2	yes	complemented
44	C₂³	—	4	2	no	—
45	D₄	—	2	4	no	—
46	D₄	—	4	2	no	—
47	C₂×C₄	—	4	2	no	—
48	C₂×C₄	—	8	2	no	—
49	C₂×C₄	—	8	2	no	—
50	C₂×C₄	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
52	C₈	—	2	1	no	—
53	C₈	—	4	2	no	—
54	C₈	—	8	2	no	—
55	C₈	C₂²×C₈	1	1	yes	characteristic
56	C₈	#131	1	1	yes	characteristic
57	C₂²	—	4	4	no	—
58	C₂²	—	8	2	no	—
59	C₂²	—	8	2	no	—
60	C₂²	#442	1	1	yes	characteristic
61	C₄	—	2	2	no	—
62	C₄	—	8	2	no	—
63	C₄	C₈×D₄	1	1	yes	characteristic
64	C₄	#444	1	1	yes	characteristic
65	C₂	—	8	4	no	—
66	C₂	#1580	1	1	yes	characteristic
67	C₂	AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#442
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂×D₁₆	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #1580	C₂	semidirect
3. #1368	C₂	semidirect
4. #1333	C₂	semidirect
5. #1313	C₂	semidirect
19. D₁₆	C₈	semidirect
32. D₈	C₂×C₈	semidirect
43. C₁₆	C₂×C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#2177	2
#1580	2
#1368	2
#1333	2
#1313	2